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{{Short description|Logarithm to the base of the mathematical constant e}}

{{Infobox mathematical function
| name = Natural logarithm
| image = Log (2).svg
| imagesize = 290px
| imagealt = Graph of part of the natural logarithm function.
| caption = Graph of part of the natural logarithm function. The function slowly grows to positive infinity as {{mvar|x}} increases, and slowly goes to negative infinity as {{mvar|x}} approaches 0 ("slowly" as compared to any [[Power law|power law]] of {{mvar|x}}).
| general_definition = <math qid=Q204037>\ln x = \log_{e} x</math>
| motivation_of_creation = Analytic proofs
| fields_of_application = Pure and applied mathematics
| domain = <math>\mathbb{R}_{> 0}</math>
| codomain = <math>\mathbb{R}</math>
| range = <math>\mathbb{R}</math>
| plusinf = +&infin;
| vr1 = {{mvar|e}}
| f1 = 1
| asymptote = <math>x = 0</math>
| root = 1
| inverse = <math>\exp x</math>
| derivative = <math>\dfrac{d}{dx}\ln x = \dfrac{1}{x} , x > 0</math>
| antiderivative = <math>\int \ln x\,dx = x \left( \ln x - 1 \right) + C</math>
}}
{{E (mathematical constant)}}

The '''natural logarithm''' of a number is its [[Logarithm|logarithm]] to the [[Base (exponentiation)|base]] of the [[Mathematical constant|mathematical constant]] [[E (mathematical constant)|{{mvar|e}}]], which is an [[Irrational number|irrational]] and [[Transcendental number|transcendental]] number approximately equal to {{math|{{val|2.718281828459}}}}.<ref>{{Cite OEIS|A001113|Decimal expansion of e}}</ref> The natural logarithm of {{mvar|x}} is generally written as {{math|ln ''x''}}, {{math|log<sub>''e''</sub> ''x''}},  or sometimes, if the base {{mvar|e}} is implicit, simply {{math|log ''x''}}.<ref>G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "''log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest''".</ref><ref>{{cite book |title=Mathematics for physical chemistry |edition=3rd |author-first=Robert G. |author-last=Mortimer |publisher=[[Company:Academic Press|Academic Press]] |date=2005 |isbn=0-12-508347-5 |page=9 |url=https://books.google.com/books?id=nGoSv5tmATsC}} [https://books.google.com/books?id=nGoSv5tmATsC&pg=PA9 Extract of page 9]</ref> Parentheses are sometimes added for clarity, giving {{math|ln(''x'')}}, {{math|log<sub>''e''</sub>(''x'')}}, or {{math|log(''x'')}}. This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm of {{mvar|x}} is the [[Exponentiation|power]] to which {{mvar|e}} would have to be raised to equal {{mvar|x}}. For example, {{math|ln 7.5}} is {{math|2.0149...}}, because {{math|1=''e''<sup>2.0149...</sup> = 7.5}}. The natural logarithm of {{mvar|e}} itself, {{math|ln ''e''}}, is {{math|1}}, because {{math|1=''e''<sup>1</sup> = ''e''}}, while the natural logarithm of {{math|1}} is {{math|0}}, since {{math|1=''e''<sup>0</sup> = 1}}.

The natural logarithm can be defined for any positive [[Real number|real number]] {{mvar|a}} as the [[Integral|area under the curve]] {{math|1=''y'' = 1/''x''}}  from {{math|1}} to {{mvar|a}}<ref name=":1">{{Cite web| last=Weisstein|first=Eric W.| title=Natural Logarithm|url=https://mathworld.wolfram.com/NaturalLogarithm.html| access-date=2020-08-29 | website=mathworld.wolfram.com | language=en}}</ref> (with the area being negative when {{math|0 < ''a'' < 1}}). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero [[Complex number|complex number]]s, although this leads to a multi-valued function: see [[Complex logarithm|complex logarithm]] for more.

The natural logarithm function, if considered as a [[Real-valued function|real-valued function]] of a positive real variable, is the [[Inverse function|inverse function]] of the [[Exponential function|exponential function]], leading to the identities:
<math display="block">\begin{align}
   e^{\ln x} &= x \qquad \text{ if } x \in \R_{+}\\
   \ln e^x   &= x \qquad \text{ if } x \in \R
 \end{align}</math>

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:<ref name=":2">{{cite web |title=Rules, Examples, & Formulas |department=Logarithm |url=https://www.britannica.com/science/logarithm|access-date=2020-08-29 |website=Encyclopedia Britannica |lang=en}}</ref>
<math display="block"> \ln( x \cdot y ) = \ln x + \ln y~.</math>

Logarithms can be defined for any positive base other than 1, not only {{mvar|e}}. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, <math>\log_b x = \ln x / \ln b = \ln x \cdot \log_b e</math>.

Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity.  For example, logarithms are used to solve for the [[Physics:Half-life|half-life]], decay constant, or unknown time in [[Exponential decay|exponential decay]] problems.  They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involving [[Compound interest|compound interest]].

==History==
{{Main|History of logarithms}}

The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and [[Biography:Alphonse Antonio de Sarasa|Alphonse Antonio de Sarasa]] before 1649.<ref>{{cite book |author-first=R.P. |author-last=Burn |date=2001 |title=Alphonse Antonio de Sarasa and Logarithms |publisher=Historia Mathematica |pages=28:1–17}}</ref> Their work involved [[Quadrature (mathematics)|quadrature]] of the [[Hyperbola|hyperbola]] with equation {{math|1=''xy'' = 1}}, by determination of the area of [[Hyperbolic sector|hyperbolic sector]]s. Their solution generated the requisite "hyperbolic logarithm" [[Function (mathematics)|function]], which had the properties now associated with the natural logarithm.

An early mention of the natural logarithm was by [[Biography:Nicholas Mercator|Nicholas Mercator]] in his work ''Logarithmotechnia'', published in 1668,<ref>{{cite web |author-first1=J. J. |author-last1=O'Connor |author-first2=E. F. |author-last2=Robertson |url=http://www-history.mcs.st-and.ac.uk/HistTopics/e.html |title=The number e |publisher=The MacTutor History of Mathematics archive |date=September 2001 |access-date=2009-02-02}}</ref> although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619.<ref name = Cajori>{{cite book |author-last=Cajori |author-first=Florian |title=A History of Mathematics |edition=5th |page=152 |publisher=AMS Bookstore |date=1991 |isbn=0-8218-2102-4 |url=https://books.google.com/books?id=mGJRjIC9fZgC}}</ref> It has been said that Speidell's logarithms were to the base {{mvar|e}}, but this is not entirely true due to complications with the values being expressed as integers.<ref name = Cajori/>{{rp|p=152}}

==Notational conventions==
The notations {{math|ln ''x''}} and {{math|log<sub>''e''</sub> ''x''}} both refer unambiguously to the natural logarithm of {{mvar|x}}, and {{math|log ''x''}} without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many [[Programming language|programming language]]s.<ref group="nb"> Including [[C (programming language)|C]], [[C++]], SAS, [[Software:MATLAB|MATLAB]], [[Software:Mathematica|Mathematica]], <!--Pascal, -->[[Fortran]], and some BASIC dialects</ref> In some other contexts such as [[Chemistry:Chemistry|chemistry]], however, {{math|log ''x''}} can be used to denote the [[Common logarithm|common (base 10) logarithm]]. It may also refer to the [[Binary logarithm|binary (base 2) logarithm]] in the context of [[Computer science|computer science]], particularly in the context of [[Time complexity|time complexity]].

==Definitions==
The natural logarithm can be defined in several equivalent ways.

===Inverse of exponential===
The most general definition is as the inverse function of <math>e^x</math>, so that <math>e^{\ln(x)} = x</math>. Because <math>e^x</math> is positive and invertible for any real input <math>x</math>, this definition of <math>\ln(x)</math> is well defined for any positive {{mvar|x}}. For the [[Complex logarithm|complex numbers]], <math>e^z</math> is not invertible, so <math>\ln(z)</math> is a [[Multivalued function|multivalued function]]. In order to make <math>\ln(z)</math> a proper, single-output [[Function (mathematics)|function]], we therefore need to restrict it to a particular [[Principal branch|principal branch]], often denoted by <math>\operatorname{Ln}(z)</math>. As the inverse function of <math>e^z</math>, <math>\ln(z)</math> can be defined by inverting the usual definition of <math>e^z</math>:
<math display="block">e^z = \lim_{n\to\infty} \left(1+\frac{z}{n}\right)^n </math>
Doing so yields:
<math display="block">\ln(z) = \lim_{n\to\infty} n\cdot (\sqrt[n]{z}-1)</math>
This definition, therefore, derives its own principal branch from the principal branch of {{mvar|n}}th roots.

===Integral definition===
[[File:Log-pole-x 1.svg|thumb|{{math|ln ''a''}} as the area of the shaded region under the curve {{math|1=''f''(''x'') = 1/''x''}} from {{math|1}} to {{mvar|a}}. If {{mvar|a}} is less than {{math|1}}, the area taken to be negative.]]
[[File:Log.gif|The area under the hyperbola satisfies the logarithm rule.  Here {{math|''A''(''s'',''t'')}} denotes the area under the hyperbola between {{mvar|s}} and {{mvar|t}}.|right|thumb]]

The natural logarithm of a positive, real number {{mvar|a}} may be defined as the [[Area|area]] under the graph of the [[Hyperbola#Rectangular hyperbola|hyperbola]] with equation {{math|1=''y'' = 1/''x''}} between {{math|1=''x'' = 1}} and {{math|1=''x'' = ''a''}}.  This is the [[Integral|integral]]<ref name=":1" />
<math display="block">\ln a = \int_1^a \frac{1}{x}\,dx.</math>
If {{mvar|a}} is in <math>(0,1)</math>, then the region has negative area, and the logarithm is negative.

This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm:<ref name=":2" />
<math display="block">\ln(ab) = \ln a + \ln b.</math>

This can be demonstrated by splitting the integral that defines {{math|ln ''ab''}} into two parts, and then making the [[Integration by substitution|variable substitution]] {{math|1=''x'' = ''at''}} (so {{math|1=''dx'' = ''a'' ''dt''}}) in the second part, as follows:
<math display="block">\begin{align}
   \ln ab = \int_1^{ab}\frac{1}{x} \, dx
   &=\int_1^a \frac{1}{x} \, dx + \int_a^{ab} \frac{1}{x} \, dx\\[5pt]
   &=\int_1^a \frac 1 x \, dx + \int_1^b \frac{1}{at} a\,dt\\[5pt]
   &=\int_1^a \frac 1 x \, dx + \int_1^b \frac{1}{t} \, dt\\[5pt]
   &= \ln a + \ln b.
 \end{align}</math>

In elementary terms, this is simply scaling by {{math|1/''a''}} in the horizontal direction and by {{mvar|a}} in the vertical direction.  Area does not change under this transformation, but the region between {{mvar|a}} and {{math|''ab''}} is reconfigured.  Because the function {{math|''a''/(''ax'')}} is equal to the function {{math|1/''x''}}, the resulting area is precisely {{math|ln ''b''}}.

The number {{mvar|e}} can then be defined to be the unique real number {{mvar|a}} such that {{math|1=ln ''a'' = 1}}.

The natural logarithm also has an improper integral representation,<ref>{{Citation |title=An improper integral representation of the natural logarithm. |url=https://www.youtube.com/watch?v=KoS5iJP5v7w |language=en |access-date=2022-09-24}}</ref> which can be derived with [[Fubini's theorem]] as follows:
<math display="block">\ln\left(x\right)=\int_1^x \frac{1}{u} du = \int_1^x \int_0^\infty e^{-tu}\ dt\ du = \int_0^\infty \int_1^x e^{-tu}\ du\ dt = \int_{0}^{\infty}\frac{e^{-t}-e^{-tx}}{t}dt</math>

==Properties==
The natural logarithm has the following mathematical properties:
* <math>\ln 1 = 0</math>
* <math>\ln e = 1</math>
* <math>\ln(xy) = \ln x + \ln y \quad \text{for }\; x > 0\;\text{and }\; y > 0</math>
* <math>\ln(x/y)= \ln x - \ln y</math>
* <math>\ln(x^y) = y \ln x \quad \text{for }\; x > 0</math>
* <math>\ln x < \ln y \quad\text{for }\; 0 < x < y</math>
* <math>\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1</math>
* <math>\lim_{\alpha \to 0} \frac{x^\alpha-1}{\alpha} = \ln x\quad \text{for }\; x > 0</math>
* <math>\frac{x-1}{x} \leq \ln x \leq x-1 \quad\text{for}\quad x > 0</math>
* <math>\ln{( 1+x^\alpha )} \leq \alpha x \quad\text{for}\quad x \ge 0\;\text{and }\; \alpha \ge 1</math>

{{Collapse top|title=Proof|width=80%|left=yes}}
The statement is true for <math>x=0</math>, and we now show that <math>\frac{d}{dx} \ln{( 1+x^\alpha )} \leq \frac{d}{dx} ( \alpha x ) </math> for all <math>x</math>, which completes the proof by the [[Fundamental theorem of calculus#First part|fundamental theorem of calculus]]. Hence, we want to show that
<math display="block">\frac{d}{dx} \ln{( 1+x^\alpha )} = \frac{\alpha x^{\alpha - 1}}{1 + x^\alpha} \leq \alpha = \frac{d}{dx} ( \alpha x ) </math>

(Note that we have not yet proved that this statement is true.) If this is true, then by multiplying the middle statement by the positive quantity <math>(1+x^\alpha) / \alpha</math> and subtracting <math>x^\alpha</math> we would obtain
<math display="block"> x^{\alpha-1} \leq x^\alpha + 1 </math>
<math display="block"> x^{\alpha-1} (1-x) \leq 1 </math>

This statement is trivially true for <math>x \ge 1</math> since the left hand side is negative or zero. For <math>0 \le x < 1</math> it is still true since both factors on the left are less than 1 (recall that <math>\alpha \ge 1</math>). Thus this last statement is true and by repeating our steps in reverse order we find that <math display="block">\frac{d}{dx} \ln{( 1+x^\alpha )} \leq \frac{d}{dx} ( \alpha x ) </math> for all <math>x</math>. This completes the proof.

An alternate proof is to observe that <math>(1+x^\alpha)\leq (1+x)^\alpha </math> under the given conditions. This can be proved, e.g., by the norm inequalities. Taking logarithms and using <math> \ln(1+x)\leq x </math> completes the proof.
{{Collapse bottom}}

== Derivative ==

The [[Derivative|derivative]] of the natural logarithm as a real-valued function on the positive reals is given by<ref name=":1" />
<math display="block">\frac{d}{dx} \ln x = \frac{1}{x}.</math>

How to establish this derivative of the natural logarithm depends on how it is defined firsthand.  If the natural logarithm is defined as the integral
<math display="block">\ln x = \int_1^x \frac{1}{t}\,dt,</math>
then the derivative immediately follows from the first part of the [[Fundamental theorem of calculus#First part|fundamental theorem of calculus]].

On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for {{math|''x'' > 0}}) can be found by using the properties of the logarithm and a definition of the exponential function.

From the definition of the number <math>e = \lim_{u\to 0}(1+u)^{1/u},</math> the exponential function can be defined as
<math display="block">e^x = \lim_{u\to 0} (1+u)^{x/u} = \lim_{h\to 0}(1 + hx)^{1/h} , </math> where <math>u=hx, h=\frac{u}{x}.</math>

The derivative can then be found from first principles.
<math display="block">\begin{align}
   \frac{d}{dx} \ln x &= \lim_{h\to 0} \frac{\ln(x+h) - \ln x}{h} \\
   &= \lim_{h\to 0}\left[ \frac{1}{h} \ln\left(\frac{x+h}{x}\right)\right] \\
   &= \lim_{h\to 0}\left[ \ln\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\right]\quad &&\text{all above for logarithmic properties}\\
   &= \ln \left[ \lim_{h\to 0}\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\right]\quad &&\text{for continuity of the logarithm} \\
   &= \ln e^{1/x} \quad &&\text{for the definition of } e^x = \lim_{h\to 0}(1 + hx)^{1/h}\\
   &= \frac{1}{x} \quad &&\text{for the definition of the ln as inverse function.}
 \end{align}</math>

Also, we have:
<math display="block">\frac{d}{dx} \ln ax = \frac{d}{dx} (\ln a + \ln x) = \frac{d}{dx} \ln a +\frac{d}{dx} \ln x = \frac{1}{x}.</math>

so, unlike its inverse function <math>e^{ax}</math>, a constant in the function doesn't alter the differential.

== Series ==

[[File:LogTay.svg|290px|thumb|right|The Taylor polynomials for {{math|ln(1 + ''x'')}} only provide accurate approximations in the range {{math|−1 < ''x'' ≤ 1}}. Beyond some {{math|''x'' > 1}}, the Taylor polynomials of higher degree are increasingly ''worse'' approximations.]]

Since the natural logarithm is undefined at 0, <math>\ln(x)</math> itself does not have a [[Maclaurin series]], unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if <math>\vert x - 1 \vert \leq 1 \text{ and } x \neq 0, </math> then<ref>{{cite web| url = http://www.math2.org/math/expansion/log.htm| title = "Logarithmic Expansions" at Math2.org}}</ref>
<math display="block">\begin{align}
   \ln x &= \int_1^x \frac{1}{t} \, dt = \int_0^{x - 1} \frac{1}{1 + u} \, du \\
   &= \int_0^{x - 1} (1 - u + u^2 - u^3 + \cdots) \, du \\
   &= (x - 1) - \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3} - \frac{(x - 1)^4}{4} + \cdots \\
   &= \sum_{k=1}^\infty \frac{(-1)^{k-1} (x-1)^k}{k}.
 \end{align}</math>

This is the [[Taylor series]] for <math>\ln x</math> around 1. A change of variables yields the [[Mercator series]]:
<math display="block">\ln(1+x)=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} x^k = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots,</math>
valid for <math>|x| \leq 1</math> and <math>x\ne -1.</math>

[[Biography:Leonhard Euler|Leonhard Euler]],<ref>[[Biography:Leonhard Euler|Leonhard Euler]], Introductio in Analysin Infinitorum. Tomus Primus. Bousquet, Lausanne 1748. Exemplum 1, p. 228; quoque in: Opera Omnia, Series Prima, Opera Mathematica, Volumen Octavum, Teubner 1922</ref> disregarding <math>x\ne -1</math>, nevertheless applied this series to <math>x=-1</math> to show that the [[Harmonic series (mathematics)|harmonic series]] equals the natural logarithm of <math>\frac{1}{1-1}</math>; that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at {{mvar|N}} is close to the logarithm of {{mvar|N}}, when {{mvar|N}} is large, with the difference converging to the [[Euler–Mascheroni constant]].

The figure is a [[Graph of a function|graph]] of {{math|ln(1 + ''x'')}} and some of its Taylor polynomials around 0. These approximations converge to the function only in the region {{math|−1 < ''x'' ≤ 1}}; outside this region, the higher-degree Taylor polynomials devolve to ''worse'' approximations for the function.

A useful special case for positive integers {{mvar|n}}, taking <math>x = \tfrac{1}{n}</math>, is:
<math display="block"> \ln \left(\frac{n + 1}{n}\right) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k n^k}  = \frac{1}{n} - \frac{1}{2 n^2} + \frac{1}{3 n^3} - \frac{1}{4 n^4} + \cdots</math>

If <math>\operatorname{Re}(x) \ge 1/2,</math> then
<math display="block">\begin{align}
   \ln (x) &= - \ln \left(\frac{1}{x}\right) = - \sum_{k=1}^\infty \frac{(-1)^{k-1} (\frac{1}{x} - 1)^k}{k} = \sum_{k=1}^\infty \frac{(x - 1)^k}{k x^k} \\
   &= \frac{x - 1}{x} + \frac{(x - 1)^2}{2 x^2} +  \frac{(x - 1)^3}{3 x^3} + \frac{(x - 1)^4}{4 x^4} + \cdots
 \end{align}</math>

Now, taking  <math>x=\tfrac{n+1}{n}</math> for positive integers {{mvar|n}}, we get:
<math display="block"> \ln \left(\frac{n + 1}{n}\right) = \sum_{k=1}^\infty \frac{1}{k (n + 1)^k} = \frac{1}{n + 1} + \frac{1}{2 (n + 1)^2} + \frac{1}{3 (n + 1)^3} + \frac{1}{4 (n + 1)^4} + \cdots</math>

If <math>\operatorname{Re}(x) \ge 0 \text{ and } x \neq 0,</math> then
<math display="block"> \ln (x) = \ln \left(\frac{2x}{2}\right) = \ln\left(\frac{1 + \frac{x - 1}{x + 1}}{1 - \frac{x - 1}{x + 1}}\right) = \ln \left(1 + \frac{x - 1}{x + 1}\right) - \ln \left(1 - \frac{x - 1}{x + 1}\right). </math>
Since
<math display="block">\begin{align}
\ln(1+y) - \ln(1-y)&= \sum^\infty_{i=1}\frac{1}{i}\left((-1)^{i-1}y^i - (-1)^{i-1}(-y)^i\right) = \sum^\infty_{i=1}\frac{y^i}{i}\left((-1)^{i-1} +1\right)  \\
&= y\sum^\infty_{i=1}\frac{y^{i-1}}{i}\left((-1)^{i-1} +1\right)\overset{i-1\to 2k}{=}\; 2y\sum^\infty_{k=0}\frac{y^{2k}}{2k+1},
\end{align}</math> 
we arrive at
<math display="block">\begin{align}
   \ln (x) &= \frac{2(x - 1)}{x + 1} \sum_{k = 0}^\infty \frac{1}{2k + 1} {\left(\frac{(x - 1)^2}{(x + 1)^2}\right)}^k \\
&= \frac{2(x - 1)}{x + 1} \left( \frac{1}{1} + \frac{1}{3} \frac{(x - 1)^2}{(x + 1)^2} + \frac{1}{5} {\left(\frac{(x - 1)^2}{(x + 1)^2}\right)}^2 + \cdots \right) .
 \end{align}</math>
Using the substitution <math>x=\tfrac{n+1}{n}</math> again for positive integers {{mvar|n}}, we get: 
<math display="block">\begin{align}
   \ln \left(\frac{n + 1}{n}\right) &= \frac{2}{2n + 1} \sum_{k=0}^\infty \frac{1}{(2k + 1) ((2n + 1)^2)^k}\\
&= 2 \left(\frac{1}{2n + 1} + \frac{1}{3 (2n + 1)^3} + \frac{1}{5 (2n + 1)^5} + \cdots \right).
\end{align}</math>

This is, by far, the fastest converging of the series described here.

The natural logarithm can also be expressed as an infinite product:<ref>{{cite web |last1=RUFFA |first1=Anthony |title=A PROCEDURE FOR GENERATING INFINITE SERIES IDENTITIES |url=https://www.emis.de/journals/HOA/IJMMS/2004/65-683653.pdf |website=International Journal of Mathematics and Mathematical Sciences |publisher=International Journal of Mathematics and Mathematical Sciences |access-date=27 February 2022}} (Page 3654, equation 2.6)</ref>
<math display="block">\ln(x)=(x-1) \prod_{k=1}^\infty \left ( \frac{2}{1+\sqrt[2^k]{x}} \right )</math>

Two examples might be:
<math display="block">\ln(2)=\left ( \frac{2}{1+\sqrt{2}} \right )\left ( \frac{2}{1+\sqrt[4]{2}} \right )\left ( \frac{2}{1+\sqrt[8]{2}} \right )\left ( \frac{2}{1+\sqrt[16]{2}} \right )...</math>
<math display="block">\pi=(2i+2)\left ( \frac{2}{1+\sqrt{i}} \right )\left ( \frac{2}{1+\sqrt[4]{i}} \right )\left ( \frac{2}{1+\sqrt[8]{i}} \right )\left ( \frac{2}{1+\sqrt[16]{i}} \right )...</math>

From this identity, we can easily get that:
<math display="block">\frac{1}{\ln(x)}=\frac{x}{x-1}-\sum_{k=1}^\infty\frac{2^{-k}x^{2^{-k}}}{1+x^{2^{-k}}}</math>

For example:
<math display="block">\frac{1}{\ln(2)} = 2-\frac{\sqrt{2}}{2+2\sqrt{2}}-\frac{\sqrt[4]{2}}{4+4\sqrt[4]{2}}-\frac{\sqrt[8]{2}}{8+8\sqrt[8]{2}}  \cdots</math>

==The natural logarithm in integration==
The natural logarithm allows simple [[Integral|integration]] of functions of the form <math>g(x) = \frac{f'(x)}{f(x)}</math>: an [[Antiderivative|antiderivative]] of {{math|''g''(''x'')}} is given by <math>\ln (|f(x)|)</math>.  This is the case because of the [[Chain rule|chain rule]] and the following fact:
<math display="block">\frac{d}{dx}\ln \left| x \right| = \frac{1}{x}, \ \  x \ne 0</math>

In other words, when integrating over an interval of the real line that does not include <math>x=0</math>, then
<math display="block">\int \frac{1}{x} \,dx = \ln|x| + C</math>
where {{mvar|C}} is an arbitrary constant of integration.<ref>For a detailed proof see for instance: George B. Thomas, Jr and Ross L. Finney, ''Calculus and Analytic Geometry'', 5th edition, Addison-Wesley 1979, Section 6-5 pages 305-306.</ref>

Likewise, when the integral is over an interval where <math>f(x) \ne 0</math>, 
:<math display="block">\int { \frac{f'(x)}{f(x)}\,dx} = \ln|f(x)| + C.</math>
For example, consider the integral of <math>\tan (x)</math> over an interval that does not include points where <math>\tan (x)</math> is infinite:
<math display="block">\int \tan x \,dx = \int \frac{\sin x}{\cos x} \,dx = -\int \frac{\frac{d}{dx} \cos x}{\cos x} \,dx = -\ln \left| \cos x \right| + C = \ln \left| \sec x \right| + C.
</math>

The natural logarithm can be integrated using [[Integration by parts|integration by parts]]:
<math display="block">\int \ln x \,dx = x \ln x - x + C.</math>

Let:
<math display="block">u = \ln x \Rightarrow du = \frac{dx}{x}</math>
<math display="block">dv = dx  \Rightarrow v = x</math>
then:
<math display="block">
\begin{align}
\int \ln x \,dx & = x \ln x - \int \frac{x}{x} \,dx \\
& = x \ln x - \int 1 \,dx \\
& = x \ln x - x + C
\end{align}
</math>

==Efficient computation==<!-- This section is linked from [[Common logarithm]] -->

For <math>\ln (x)</math> where {{math|''x'' > 1}}, the closer the value of {{mvar|x}} is to 1, the faster the rate of convergence of its Taylor series centered at 1. The identities associated with the logarithm can be leveraged to exploit this:
<math display="block">\begin{align}
\ln 123.456 &= \ln(1.23456 \cdot 10^2)\\
&= \ln 1.23456 + \ln(10^2)\\
&= \ln 1.23456 + 2 \ln 10\\
&\approx \ln 1.23456 + 2 \cdot 2.3025851.
\end{align}</math>

Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above.

===Natural logarithm of 10===
The natural logarithm of 10, approximately equal to {{math|{{val|2.30258509}}}},<ref>{{Cite OEIS|A002392|Decimal expansion of natural logarithm of 10}}</ref> plays a role for example in the computation of natural logarithms of numbers represented in [[Scientific notation|scientific notation]], as a mantissa multiplied by a power of 10:
<math display="block">\ln(a\cdot 10^n) = \ln a + n \ln 10.</math>

This means that one can effectively calculate the logarithms of numbers with very large or very small [[Magnitude (mathematics)|magnitude]] using the logarithms of a relatively small set of decimals in the range {{math|[1, 10)}}.

==={{anchor|lnp1}}High precision===
To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if {{mvar|x}} is near 1, a good alternative is to use [[Halley's method]] or [[Newton's method]] to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of {{mvar|y}} to give <math>\exp(y)-x=0</math> using Halley's method, or equivalently to give <math>\exp \Bigl(\frac{y}{2}\Bigr) -x \exp \Bigl(-\frac{y}{2}\Bigr)=0</math> using Newton's method, the iteration simplifies to
<math display="block"> y_{n+1} = y_n + 2 \cdot \frac{ x - \exp ( y_n ) }{ x + \exp ( y_n ) } </math>
which has cubic convergence to <math>\ln (x)</math>.

Another alternative for extremely high precision calculation is the formula<ref>{{cite journal |author-first1=T. |author-last1=Sasaki |author-first2=Y. |author-last2=Kanada |title=Practically fast multiple-precision evaluation of log(x) |journal=Journal of Information Processing |volume=5 |issue=4 |pages=247–250 |date=1982 |url=http://ci.nii.ac.jp/naid/110002673332 |access-date=2011-03-30}}</ref><ref>{{cite journal |author-first1=Timm |author-last1=Ahrendt |journal=Stacs 99 |series=Lecture Notes in Computer Science |doi=10.1007/3-540-49116-3_28 |volume=1564 |date=1999 |pages=302–312 |title=Fast Computations of the Exponential Function |isbn=978-3-540-65691-3}}</ref>
<math display="block">\ln x \approx \frac{\pi}{2 M(1,4/s)} - m \ln 2,</math>
where {{mvar|M}} denotes the arithmetic-geometric mean of 1 and {{math|4/''s''}}, and
<math display="block">s = x 2^m > 2^{p/2},</math>
with {{mvar|m}} chosen so that {{mvar|p}} bits of precision is attained. (For most purposes, the value of 8 for {{mvar|m}} is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants <math>\ln 2</math> and [[Pi|{{pi}}]] can be pre-computed to the desired precision using any of several known quickly converging series.) Or, the following formula can be used:
<math display="block">\ln x = \frac{\pi}{M\left(\theta_2^2(1/x),\theta_3^2(1/x)\right)},\quad x\in (1,\infty)</math>

where
<math display="block">
\theta_2(x) = \sum_{n\in\Z} x^{(n+1/2)^2},
 \quad
\theta_3(x) = \sum_{n\in\Z} x^{n^2}
</math>
are the [[Theta function#Auxiliary functions|Jacobi theta functions]].<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} page 225</ref>

Based on a proposal by [[Biography:William Kahan|William Kahan]] and first implemented in the [[Company:Hewlett-Packard|Hewlett-Packard]] HP-41C calculator in 1979 (referred to under "LN1" in the display, only), some calculators, [[Operating system|operating system]]s (for example Berkeley UNIX 4.3BSD<ref name="Beebe_2017"/>), [[Software:Computer algebra system|computer algebra system]]s and programming languages (for example [[C99]]<ref name="Beebe_2002"/>) provide a special '''natural logarithm plus 1''' function, alternatively named '''LNP1''',<ref name="HP48_AUR">{{cite book |title=HP&nbsp;48G Series – Advanced User's Reference Manual (AUR) |publisher=[[Company:Hewlett-Packard|Hewlett-Packard]] |edition=4 |date=December 1994 |id=HP 00048-90136, 0-88698-01574-2 |orig-year=1993<!-- edition 1 (1993-07) --> |url=http://www.hpcalc.org/details.php?id=6036 |access-date=2015-09-06}}</ref><ref name="HP50_AUR">{{cite book |title=HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR) |publisher=[[Company:Hewlett-Packard|Hewlett-Packard]] |edition=2 |date=2009-07-14 |orig-year=2005<!-- first published: Edition 1 (2005–09) --> |id=HP F2228-90010 |url=http://www.hpcalc.org/details.php?id=7141 | access-date=2015-10-10}} [http://holyjoe.net/hp/HP_50g_AUR_v2_English_searchable.pdf Searchable PDF]</ref> or '''log1p'''<ref name="Beebe_2002"/> to give more accurate results for logarithms close to zero by passing arguments {{mvar|x}}, also close to zero, to a function {{math|log1p(''x'')}}, which returns the value {{math|ln(1+''x'')}}, instead of passing a value {{mvar|y}} close to 1 to a function returning {{math|ln(''y'')}}.<ref name="Beebe_2002">{{cite web |title=Computation of expm1 = exp(x)−1 |author-first=Nelson H. F. |author-last=Beebe |publisher=Department of Mathematics, Center for Scientific Computing, [[Organization:University of Utah|University of Utah]] |location=Salt Lake City, Utah, USA |date=2002-07-09 |version=1.00 |url=http://www.math.utah.edu/~beebe/reports/expm1.pdf |access-date=2015-11-02}}</ref><ref name="HP48_AUR"/><ref name="HP50_AUR"/> The function {{math|log1p}} avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the natural logarithm. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers.<ref name="HP48_AUR"/><ref name="HP50_AUR"/>

In addition to base {{mvar|e}}, the [[IEEE 754-2008]] standard defines similar logarithmic functions near 1 for [[Binary logarithm|binary]] and [[Decimal logarithm|decimal logarithm]]s: {{math|log<sub>2</sub>(1 + ''x'')}} and {{math|log<sub>10</sub>(1 + ''x'')}}.

Similar inverse functions named "expm1",<ref name="Beebe_2002"/> "expm"<ref name="HP48_AUR"/><ref name="HP50_AUR"/> or "exp1m" exist as well, all with the meaning of {{math|1=expm1(''x'') = exp(''x'') − 1}}.<ref group="nb" name="Alternative_funcs">For a similar approach to reduce [[Round-off error|round-off error]]s of calculations for certain input values see trigonometric functions like [[Versine|versine]], [[Vercosine|vercosine]], [[Coversine|coversine]], [[Covercosine|covercosine]], [[Haversine|haversine]], [[Havercosine|havercosine]], [[Hacoversine|hacoversine]], [[Hacovercosine|hacovercosine]], [[Exsecant|exsecant]] and [[Excosecant|excosecant]].</ref>

An identity in terms of the inverse hyperbolic tangent,
<math display="block">\mathrm{log1p}(x) = \log(1+x) = 2 ~ \mathrm{artanh}\left(\frac{x}{2+x}\right)\,,</math>
gives a high precision value for small values of {{mvar|x}} on systems that do not implement {{math|log1p(''x'')}}.

===Computational complexity===
{{main|Computational complexity of mathematical operations}}
The [[Computational complexity theory|computational complexity]] of computing the natural logarithm using the [[Arithmetic–geometric mean|arithmetic-geometric mean]] (for both of the above methods) is <math>\text{O}\bigl(M(n) \ln n \bigr)</math>. Here, {{mvar|n}} is the number of digits of precision at which the natural logarithm is to be evaluated, and {{math|''M''(''n'')}} is the computational complexity of multiplying two {{mvar|n}}-digit numbers.

==Continued fractions==
While no simple [[Continued fraction|continued fraction]]s are available, several [[Generalized continued fraction|generalized continued fraction]]s exist, including:
<math display="block">
\begin{align}
\ln(1+x) & =\frac{x^1}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots \\[5pt]
& = \cfrac{x}{1-0x+\cfrac{1^2x}{2-1x+\cfrac{2^2x}{3-2x+\cfrac{3^2x}{4-3x+\cfrac{4^2x}{5-4x+\ddots}}}}}
\end{align}
</math>
<math display="block">
\begin{align}
\ln\left(1+\frac{x}{y}\right) & = \cfrac{x} {y+\cfrac{1x} {2+\cfrac{1x} {3y+\cfrac{2x} {2+\cfrac{2x} {5y+\cfrac{3x} {2+\ddots}}}}}} \\[5pt]
& = \cfrac{2x} {2y+x-\cfrac{(1x)^2} {3(2y+x)-\cfrac{(2x)^2} {5(2y+x)-\cfrac{(3x)^2} {7(2y+x)-\ddots}}}}
\end{align}
</math>

These continued fractions—particularly the last—converge rapidly for values close to&nbsp;1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence.

For example, since 2 = 1.25<sup>3</sup> × 1.024, the [[Natural logarithm of 2|natural logarithm of 2]] can be computed as:
<math display="block">
\begin{align}
\ln 2 & = 3 \ln\left(1+\frac{1}{4}\right) + \ln\left(1+\frac{3}{125}\right) \\[8pt]
& = \cfrac{6} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
+ \cfrac{6} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
\end{align}
</math>

Furthermore, since 10 = 1.25<sup>10</sup> × 1.024<sup>3</sup>, even the natural logarithm of 10 can be computed similarly as:
<math display="block">
\begin{align}
\ln 10 & = 10 \ln\left(1+\frac{1}{4}\right) + 3\ln\left(1+\frac{3}{125}\right) \\[10pt]
& = \cfrac{20} {9-\cfrac{1^2} {27-\cfrac{2^2} {45-\cfrac{3^2} {63-\ddots}}}}
+ \cfrac{18} {253-\cfrac{3^2} {759-\cfrac{6^2} {1265-\cfrac{9^2} {1771-\ddots}}}}.
\end{align}
</math>
The reciprocal of the natural logarithm can be also written in this way:
<math display="block">\frac {1}{\ln(x)} = \frac {2x}{x^2-1}\sqrt{\frac {1}{2}+\frac {x^2+1}{4x}}\sqrt{\frac {1}{2}+\frac {1}{2}\sqrt{\frac {1}{2}+\frac {x^2+1}{4x}}}\ldots</math>

For example:
<math display="block">\frac {1}{\ln(2)} = \frac {4}{3}\sqrt{\frac {1}{2} + \frac {5}{8}} \sqrt{\frac {1}{2} + \frac {1}{2} \sqrt{\frac {1}{2} +\frac {5}{8}}} \ldots</math>

==Complex logarithms==
{{Main|Complex logarithm}}
The exponential function can be extended to a function which gives a [[Complex number|complex number]] as {{math|''e''<sup>''z''</sup>}} for any arbitrary complex number {{mvar|z}}; simply use the infinite series with {{mvar|x}}=z complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no {{mvar|x}} has {{math|1=''e''<sup>''x''</sup> = 0}}; and it turns out that {{math|1=''e''<sup>2''iπ''</sup> = 1 = ''e''<sup>0</sup>}}. Since the multiplicative property still works for the complex exponential function, {{math|1=''e''<sup>''z''</sup> = ''e''<sup>''z''+2''kiπ''</sup>}}, for all complex {{mvar|z}} and integers&nbsp;{{mvar|k}}.

So the logarithm cannot be defined for the whole [[Complex plane|complex plane]], and even then it is multi-valued—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of {{math|2''iπ''}} at will. The complex logarithm can only be single-valued on the [[Complex plane#Cutting the plane|cut plane]].  For example, {{math|ln ''i'' {{=}} {{sfrac|''iπ''|2}}}} or {{math|{{sfrac|5''iπ''|2}}}} or {{math|-{{sfrac|3''iπ''|2}}}}, etc.; and although {{math|''i''<sup>4</sup> {{=}} 1, 4 ln ''i''}} can be defined as {{math|2''iπ''}}, or {{math|10''iπ''}} or {{math|−6''iπ''}}, and so on.

<gallery mode="packed" caption="Plots of the natural logarithm function on the complex plane ([[Principal branch|principal branch]])">
Image:NaturalLogarithmRe.png|{{math|''z'' {{=}} Re(ln(''x'' + ''yi''))}} 
Image:NaturalLogarithmImAbs.png|{{math|''z'' {{=}} {{abs|(Im(ln(''x'' + ''yi'')))}}}} 
Image:NaturalLogarithmAbs.png|{{math|''z'' {{=}} {{abs|(ln(''x'' + ''yi''))}}}}
Image:NaturalLogarithmAll.png| Superposition of the previous three graphs
</gallery>

==See also==
* Approximating natural exponents (log base e)
* [[Iterated logarithm]]
* [[Napierian logarithm]]
* [[List of logarithmic identities]]
* [[Logarithm of a matrix]]
* [[Exponential map (Lie theory)#Logarithmic coordinates|Logarithmic coordinates]] of an element of a Lie group.
* [[Logarithmic differentiation]]
* [[Logarithmic integral function]]
* [[Biography:Nicholas Mercator|Nicholas Mercator]] – first to use the term natural logarithm
* [[Polylogarithm]]
* [[Von Mangoldt function]]

==Notes==
{{Reflist|group="nb"}}

==References==
{{Reflist|refs=
<ref name="Beebe_2017">{{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter 10.4. Logarithm near one |date=2017-08-22 |location=Salt Lake City, UT, USA |publisher=Springer International Publishing AG |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |pages=290–292 |s2cid=30244721 |quote=In 1987, Berkeley UNIX 4.3BSD introduced the log1p() function}}</ref>
}}

{{Calculus topics}}

[[Category:Logarithms]]
[[Category:Elementary special functions]]
[[Category:E (mathematical constant)]]
[[Category:Unary operations]]

de:Logarithmus#Natürlicher Logarithmus

{{Sourceattribution|Natural logarithm}}